Basics of Algebra: Part I

by Oren Lahav

Algebra, part I- Numbers, Variables, Operations, Expressions, Relations

This series of lessons is designed to help you learn, or review, the fundamentals of algebra. We start off with the very basics- numbers, variables, and operations.

Algebra is one of the main branches of mathematics. It concerns studying number structures and relations. Let's start with the very first thing that math involves- numbers.

Numbers- these are the symbols we use to do math. For example, 3, -289, 0, five.

There are actually various types of numbers around. They include:

A. Real numbers This group, denoted by a lareg R, represents the large portion of numbers you've ever seen. Every numberyou can place on a number line is real.

B. Natural numbers This group, generally denoted by an N, is the group of all positive integers, like 7, 18, or 219499234.

C. Integers This group is denoted by a Z, and it represents all integers, including the natural numbers, 0, and negative multiples of all natural numbers like -5 and -2439.

D. Rationals This group is Q, and it represents all nice numbers that can be expressed as a ratio of two integers. 1, 1/3, 0.4 and -6/7 are all rational numbers, while $\sqrt{2}$ or $\pi$ are irrational numbers.

Numbers are cool. You may ask yourself, why do I need all of these different types of numbers? Are there unreal numbers floating around, and if so, what are they used for? Why do we need irrationals, can't all numbers just be nice?

Questions such as this will be answered in a future part of the series.

For now, let's introduce the next big step- variables:

Variables are non-numerical representations of numbers. They're usually letters, like X or Y. They can also be Greek letters like alpha, or crazy stuff like @. In actuality, variables can be anything you can think of, like a pickle or a shoe.

So what are variables useful for? We use variables to represent unknown numbers in mathematical expressions. Before we can move on to mathematical expressions, however, we have to take a look at one major thing first.

Introducing- operations:

Operations are basic actions you can perform with your numbers and variables.

The basic operations are:

bq. A. Addition: for example, 1+2=3

bq. B. Multiplication:for example, 2 * 3=6

There are other minor variations of these operations:

bq. C. Subtraction, adding a negative: 5+(-2)=5-2=3

bq. D. Division, multiplying by an inverse: $4 * (\frac{1}{2})=\frac{4}{2}=2$ .

Cool.

Now, let's put it all together:

Sounds like a joke, right? Ok, here goes: 5x+2 . Wait… now what?
This is a mathematical expression. Expressions that involve one or more variables and some constants are called polynomials%. $\frac{6a ^ 7}{11}+\frac{4a}{3}+326$ is another polynomial. There are lots of polynomials around, but what are they good for?

On their own, polynomials aren't good for much. But they become a whole lot more useful when we add this:

Relations between mathematical variables

Relations in math, you may be surprised to know, have nothing to do with my various ex-girlfriends. These mathematical symbols include "=", ">", "<", etc. They allow us to compare mathematical expressions.

Here's a super easy example: x=4 . Here's a slightly more complicated example: 2x=8 . You may already realize these are equivalent, i.e. they mean the same thing. How do I know that? Simple, because mathematical expressions can be simplified. For example, say 5x+3=23 . We can subtract 3 from both sides to get 5x=20 , then divide by 5 to get x=4 . This process is called Solving mathematical equations (with 1 unknown).